Assemble the cake: Preheat oven to 350°F. Thoroughly grease and flour three 9-inch round pans. In a medium bowl, whisk together the flour, baking powder and salt. In another large bowl, whisk together the sugar, yogurt, and lemon juice. Slowly whisk in the oil, followed by the eggs one at a time, and then the vanilla. Sift and fold in the dry mixture just until combined. Distribute evenly in the pans (if you want to get a really even distribution, you can actually measure the batter into each pan) and bake until a toothpick comes out clean, about 20 minutes. Cool completely.

Assemble the cream: Combine the cream, sugar, lemon zest, and lemon juice in a large bowl and whip. Refrigerate.

Now, your fruit: In a small saucepan over medium heat, add 1 tablespoon lemon juice, 1 tablespoon sugar, and 1 cup of berries. Simmer over low heat until almost jam-like. Refrigerate.

Once everything is cooled completely, put it all together: Put one layer of your cake onto a cake stand or plate. Top with a third of the whipped cream, a third of the compote, a third of the berries, and a sprinkle of pistachios and thyme leaves. Repeat for the next two layers.